Legendre Polynomial Expansion of the Electron Boltzmann Equation Applied to the Discharge in Argon

by Sosov, Yuriy

Abstract (Summary)

The main effort of the present dissertation is to establish a framework for construction of the numerical solution of the system of partial differential equations for the coefficients in the N-term expansion of the solution of the Boltzmann equation in Legendre polynomials, also known as the PN approximation of the Boltzmann equation. The key feature of the discussed solution is the presence of multiple waves moving in opposite directions in both velocity and spatial domains, which requires transformation of the expansion coefficients to characteristic variables and a directional treatment (up/down winding) of their velocity and spatial derivatives. After the presence of oppositely directed waves in the general solution is recognized, the boundary conditions at the origin of velocity space are formulated in terms of the arriving and reflected waves, and the meaning of the characteristic variables is determined, then the construction proceeds employing the standard technique of operator splitting. Special effort is made to insure numerically exact particle conservation in treatment of the advection and scattering processes. The constructed numerical routine has been successfully coupled with a solver for the Poisson equation in a self-consistent model of plasma discharge in argon for a two parallel-plate bare electrode geometry. The results of this numerical experiment were presented at the workshop on "Nonlocal, Collisionless Electron Transport in Plasmas" held at Plasma Physics Laboratory of Princeton University on August 2-4, 2005.